Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the capacity of sets of divergence associated with the spherical partial integral operator

Author: Emmanuel Montini
Journal: Trans. Amer. Math. Soc. 355 (2003), 1415-1441
MSC (2000): Primary 42B05, 31B15
Published electronically: November 14, 2002
MathSciNet review: 1946398
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this article, we study the pointwise convergence of the spherical partial integral operator $S_Rf(x)=\int_{B(0,R)} \hat{f} (y) e^{2\pi ix\cdot y}dy$ when it is applied to functions with a certain amount of smoothness. In particular, for $f\in \mathcal{L}_{\alpha}^p(\mathbb{R} ^n)$, $\tfrac{n-1}{2} <\alpha\leq\tfrac{n}{p}$, $2\leq p<\tfrac{2n}{n-1}$, we prove that $S_Rf(x)\to G_{\alpha} *g(x)$ $C_{\alpha,p}$-quasieverywhere on $\mathbb{R} ^n$, where $g\in L^p({\mathbb{R} }^n )$ is such that $f=G_{\alpha}*g$ almost everywhere. A weaker version of this result in the range $0<\alpha\leq\tfrac{n-1}{2}$ as well as some related localisation principles are also obtained. For $1\leq p<2-\tfrac{1}{n}$ and $0\leq\alpha <\tfrac{(2-p)n-1}{2p}$, we construct a function $f\in\mathcal{L}_\alpha^p(\mathbb{R} ^n)$ such that $S_Rf(x)$ diverges everywhere.

References [Enhancements On Off] (What's this?)

  • 1. D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften No. 314, Springer-Verlag, Berlin, 1996. MR 97j:46024
  • 2. A. Beurling, Ensembles exceptionnels, Acta Math. 72 (1940), pp. 1-13.MR 1:226a
  • 3. A. Carbery, The Boundedness of the Maximal Bochner-Riesz Operator on $L^4({\mathbb{R} }^2)$, Duke Math. J. 50 (1983), 409-416. MR 84m:42025
  • 4. A. Carbery, J. L. Rubio de Francia and L. Vega, Almost Everywhere Summability of Fourier Integrals, J. London Math. Soc. (2) 38 (1988), pp. 513-524. MR 90c:42033
  • 5. A. Carbery and F. Soria, Almost Everywhere Convergence of Fourier Integrals for Functions in Sobolev Spaces and an $L^2$-Localisation Principle, Rev. Mat. Iberoamericana 4 (1988), pp. 319-337. MR 91d:42015
  • 6. A. Carbery and F. Soria, Pointwise Fourier Inversion and Localisation in ${\mathbb{R} }^n$, J. Fourier Anal. Appl. 3 (1997), Special Issue, pp. 847-858. MR 99c:42018
  • 7. A. Carbery and F. Soria, personal communication (MSRI (Berkeley, USA), 1997).
  • 8. L. Carleson, On Convergence and Growth of Partial Sums of Fourier Series, Acta Math. 116 (1966), pp. 133-157. MR 33:7774
  • 9. L. Carleson, Selected Problems on Exceptional Sets, Van Nostrand Mathematical Studies No.13, Van Nostrand, Toronto, 1967. MR 37:1576
  • 10. M. Christ, On Almost Everywhere Convergence of Bochner-Riesz Means in Higher Dimensions, Proc. Amer. Math. Soc. 95 (1985), pp. 16-20. MR 87c:42020
  • 11. C. Fefferman, A Note on Spherical Summation Multipliers, Israël J. Math. 15 (1973), pp. 44-52. MR 47:9160
  • 12. R. A. Hunt, On the Convergence of Fourier Series, Proc. of the Conf. on Orthogonal Expansions and their Continuous Analogues, Southern Illinois Univ. Press, 1968, pp. 235-255. MR 38:6296
  • 13. V. A. Il'in, The Problems of Localization and Convergence of Fourier Series with Respect to the Fundamental Systems of Functions of the Laplace Operator, Russian Math. Surv. 23 No.2 (1968), pp. 59-116. MR 36:6870
  • 14. J.-P. Kahane and R. Salem, Ensembles parfaits et séries trigonométriques, Actualités Sci. Indust. 1301, Hermann, Paris, 1963. MR 28:3279
  • 15. Y. Kanjin, Convergence and Divergence Almost Everywhere of Spherical Means for Radial Functions, Proc. Amer. Math. Soc. 103 (1988), pp. 1063-1069. MR 89i:42030
  • 16. C. E. Kenig and P. A. Tomas, Maximal Operators Defined by Fourier Multipliers, Studia Math. 68 (1980), pp. 79-83. MR 82c:42016
  • 17. P. Mattila, Spherical Averages of Fourier Transforms of Measures with Finite Energy, Mathematika 34 (1987), 207-228. MR 90a:42009
  • 18. C. Meaney and E. Prestini, On Convergence of some Integral Transforms, Proc. Centre Math. Appl. Austral. Nat. Univ. 29, 1992, pp. 145-162. MR 93i:42007
  • 19. E. Montini, Questions Related with the Inversion of the Fourier Transform in Dimensions Greater than 1 for Functions in ${\mathcal{L}}_{\alpha}^p$, Ph.D. Thesis, Univ. of Edinburgh (U.K.), 1998.
  • 20. E. Prestini, Almost Everywhere Convergence of Spherical Partial Sums for Radial Functions, Monatsh. Math. 105 (1988), pp. 207-216. MR 89g:42028
  • 21. F. E. Relton, Applied Bessel Functions, Blackie & Son Ltd., London, 1946. MR 9:584b
  • 22. R. Salem and A. Zygmund, Capacity of Sets and Fourier Series, Trans. Amer. Math. Soc. 59 (1946), pp. 23-41. MR 7:434h
  • 23. P. Sjölin, Two Theorems on Convergence of Fourier Integrals and Fourier Series, in Approximation and Function Spaces, Banach Center Publ. vol.22, PWN-Polish Scientific Publ., 1989, pp. 413-426. MR 92e:42004
  • 24. P. Sjölin, Estimates of Spherical Averages of Fourier Transforms and Dimensions of Sets, Mathematika 40 (1993), pp. 322-330. MR 95f:28007
  • 25. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. No.30, Princeton Univ. Press, Princeton, NJ, 1970. MR 44:7280
  • 26. E. M. Stein, Harmonic Analysis - Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Math. Ser. No.43, Princeton Univ. Press, Princeton, NJ, 1993. MR 95c:42002
  • 27. E. M. Stein and G. Weiss, An Introduction to Fourier Analysis in Euclidean Spaces, Princeton Math. Ser. No.32, Princeton Univ. Press, Princeton, NJ, 1971. MR 46:4102
  • 28. T. Tao, The Weak-Type Endpoint Bochner-Riesz Conjecture and Related Topics, Indiana J. of Math. 47 (1998), pp. 1097-1124. MR 2000a:42024
  • 29. G. N. Watson, A Treatise on the Theory of Bessel Functions, Second Ed., Cambridge Univ. Press, Cambridge, 1944. MR 6:64a

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42B05, 31B15

Retrieve articles in all journals with MSC (2000): 42B05, 31B15

Additional Information

Emmanuel Montini
Affiliation: Department of Mathematics and Statistics, University of Edinburgh, J.C.M.B., King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom
Address at time of publication: NEKS Technologies Inc., 230 rue Bernard-Belleau, Bureau 221, Laval (Québec) H7V 4A9, Canada

Received by editor(s): August 31, 2000
Published electronically: November 14, 2002
Additional Notes: The author was supported in part by a Commonwealth Academic Staff Fellowship (CA0355)
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society